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equivariant cell structure on 2-tori – section

The following shows for the symmorphic 2D crystallographic groups (wallpaper groups) $\mathbb{Z}^2 \rtimes G \subset Iso(2)$ the $G$-CW complex structure on the resulting tori $T^2 \coloneqq \mathbb{R}^2/\mathbb{Z}^2$.

The point groups $G$ arising are the cyclic groups $\mathbb{Z}_{/1}=$$1$, $\mathbb{Z}_{/2}$, $\mathbb{Z}_{/3}$, $\mathbb{Z}_{/4}$, $\mathbb{Z}_{/6}$ and the dihedral groups $Dih_1$, $Dih_2$, $Dih_3$, $Dih_4$, and $Dih_6$ (making 10 distinct point groups $G$, but the first three of the latter come with two inequivalent actions each).

p1TorusCellStructure-20250722.png

pmTorusCellStructure-20250722.png

cmTorusCellStructure-20250722.png

p2TorusCellStructure-20250722.png

pmmTorusCellStructure-20250726.png

cmmTorusCellStructure-20250722.png

p3TorusCellStructure-20250722.png

p31mTorusCellStructure-20250722.png

p3m1TorusCellStructure-20250722.png

p4TorusCellStructure-20250722.png

p4mTorusCellStructure-20250722.png

p4gTorusCellStructure-20250802.png?

p6TorusCellStructure-20250722.png

p6mTorusCellStructure-20250722.png

pgTorusCellStructure-20250726b.png

pmgTorusCellStructure-20250730.png

pggTorusCellStructure-20250731.png